# Milieux Continus-Continuous Media-S2

Milieux Continus / Continuous Media

Contact: A. Pocheau & J. Deschamps

Motivations: The course addresses the dynamics of deformable continuous media, especially fluid media, including their kinematic description and their transport and exchange phenomena. It aims at identifying the dynamical regimes, the specific features of flows and their main implications.

The tools for describing the continuous media will include strains, kinematics (strain rate, vorticity, shear) and geometry (streamlines and stream functions, trajectories). Then advective or diffusive fluxes of scalar or vectorial quantities (impulsion) will be introduced and modelled by the Fourier law or the Fick law. The dynamical budgets of these quantities in eulerian or lagrangian control volume will provide constitutive differential equations (Navier-Stokes, Fourier, Fick).

Analysis of the dynamics will begin by geometrical and dynamical similarity and yield essential non-dimensional numbers (Reynolds, Péclet). The viscous Stokes regime will be addressed in Hele-Shaw cells, Poiseuille flows, boundary layers and porous media in link with reversibility/irreversibility concepts.

The inviscid fluid regime will be addressed by the Euler equation, the Bernoulli relation, and potential flow. Conformal mappings will be applied to deduce flows around bodies and the resulting forces.

The vorticity dynamics will be considered in analogy with electromagnetism and will yield strain/diffusion dynamics, the Kelvin theorem and the d’Alembert paradox.

This course provides a valuable preparation to the S3-courses « Dynamical System and Non-Linear Physics», « Soft Matter » and to the « concours de l’agrégation de physique-chimie, option physique ».

I] Foundation

• description of continuous media : deformation, kinematics, geometry.
• modelling of exchange and transport phenomena (flux of scalar/vectorial quantity, stress, viscosity)
• budget equations : Euler/Lagrange formulation, Navier-Stokes equation, Fourier/Fick equations, incompressibility.

II] Developments

• geometrical/dynamical similarity.
• Stokes regime, perfect fluid, potential flows.
• vorticity, planar boundary layers, applications.